Rigidity of Einstein metrics as critical points of quadratic curvature functionals on closed manifolds

TL;DR

This paper establishes rigidity results for Einstein metrics on closed manifolds by analyzing their critical points of quadratic curvature functionals, involving inequalities related to curvature and invariants.

Contribution

It introduces new rigidity theorems for Einstein metrics based on quadratic curvature functionals and curvature invariants, expanding understanding of geometric stability.

Findings

Rigidity results for Einstein metrics as critical points of quadratic curvature functionals.

Conditions involving Weyl and trace-less Ricci curvature lead to new geometric constraints.

Results connect curvature inequalities with the Yamabe invariant for Einstein metrics.

Abstract

In this paper, we prove some rigidity results for the Einstein metrics as the critical points of a family of known quadratic curvature functionals on closed manifolds, characterized by some point-wise inequalities. Moreover, we also provide a few rigidity results that involve the Weyl curvature, the trace-less Ricci curvature and the Yamabe invariant, accordingly.